The definitions of the combinatorial and normalised Laplacians can be naturally extended to edge-weighted graphs \((G,\varepsilon )\) , and these have been studied by several authors, including by Kirchhoff in his original paper [1]}, where the edge weights represent the conductances of wires. In contrast, graph Laplacians with vertex weights are not so commonly studied, and there exist competing definitions for such a matrix [2]}, [3]}, [4]}. Vertex-weighted graphs appear in the work of Noble and Welsh [5]}, in which they define a polynomial associated to integer vertex-weighted graphs satisfying a deletion-contraction rule, motivated by the study of Vassiliev invariants of knots [6]}. Vertex weights also arise in graph theory applications; when studying independent sets in job scheduling, the weight of a vertex can be used to represent the relative reward for the completion of each job.
